When a ring is a polynomial ring?

In the paper (2.11) the authors show that if k∗ is a separable algebraic extension of k and x1,x2,…,xn are indeterminates over k∗ and a normal one dimensional ring A with k⊂A⊂k∗[x1,x2,…,xn] then A has the form k′[t] where k′ is the algebraic closer of k in A. The above is a very strict sufficient … Read more

Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms

I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite k-algebras (k is a (preferably, but not necessarily finite) field) with one or more of the following properties: 1) the Auslander-Reiten quiver contains two cylinders (so, there are periodic modules), but also many non-periodic modules 2) the Auslander-Reiten quiver contains … Read more

Equivariant sheaves over affine schemes

Let k be a field, let G be a linear algebraic group over k and let A be a commutative k-algebra which is acted on by G. We say that an A-module M is a (G,A)-module if it satisfies the following two properties: 1) M is a rational G-module (over k) 2) The multiplication map … Read more

A right adjoint to the truncated Witt functor?

For any ring A, let wEtA be the category of weakly etale A-algebras ; it is a cocomplete category. By a theorem of Van der Kallen, the truncated Witt vector functor Wr:wEtA⟶wEtWr(A) is well-defined and commutes with all colimits. Can one explicit a right adjoint to Wr ? Answer AttributionSource : Link , Question Author … Read more

Does a polynomial system with precisely e solutions have a Groebner basis of degree bounded by e?

Let k be a field and let R=k[X1,…,Xn] be a polynomial ring. Let F⊂R be a finite subset generating a radical ideal I with precisely e solutions over an algebraic closure of k. Is there a monomial order on R with a Groebner basis of degree ≤e for I? Answer AttributionSource : Link , Question … Read more

When is a given polynomial a square of another polynomial?

I meet a problem in which I hope to show a special polynomial is not a square of another polynomial. More precisely, let’s consider the polynomial f(x):=1−x+2bxn−2bxn+1−b2x2n−1+2b2x2n−b2x2n+1−2bx3n−1+2bx3n−x4n−1+x4n∈k[x], where k is a field of characteristic p>0, n>2 is an integer, and b∈k with b≠0,1,−1. Indeed, in the context I meet, the field k is just the … Read more

polynomials satisfying the Plücker relation

Let $S_{12}$, $S_{13}$, $S_{14}$, $S_{23}$, $S_{24}$, $S_{34}$ be complex homogeneous polynomials in 4 variables satisfying the Plücker relation : $$S_{12}S_{34}-S_{13}S_{24}+S_{14}S_{23}=0 .$$ Suppose also that they don’t have any common non trivial zero. Let $d_{ij}=\deg(S_{ij})$. Let $d$ be the integer $$d=d_{12}+d_{34}=d_{13}+d_{24}=d_{14}+d_{23}.$$ By an indirect way (I used this to construct vector bundles on $\mathbb{P}_3$), I find … Read more

Methods to check if an ideal of a polynomial ring is prime

Fix $\ell \geq 3$, $r \geq 2$ and $1 \leq k \leq \ell – 1$ and $z_1, \ldots, z_\ell \in \mathbb{C}$ with $z_i \neq 0$ for all $i$ and $z_i \neq z_j$ for all $i \neq j$. Now consider the (irreducible, non-homogeneous) polynomials $q_i = z_{\ell}x_i^r – z_ix_{\ell}^r – (z_{\ell} – z_i)$ for $1 \leq … Read more

Chevalley-Warning for finite rings: the degree of a non-polynomial

\def\F{\mathbb F} \def\Z{\mathbb Z} One reason that Chevalley-Warning theorem is that amazingly useful is the fact that for a finite field \F, any function from \F^n to \F is a polynomial. This fails to hold in general for finite rings; say, it is easy to see that if R is a finite (commutative) ring but … Read more